2003

AbstractSimple jerk systems are very useful for combining analytical computations and dynamical analysis in phase space. This is particularly relevant since there is still no direct link between the algebraic structure of ordinary differential equations and the topology of the chaotic attractors which they generate. In this paper, particular analytical solutions are identified for three simple chaotic flows. It is shown that these solutions have varying effects on the bifurcation diagrams. Moreover, a feedback circuit analysis is used to exhibit the similarities between the three simple systems. Such analysis also exhibits the relevant role of double nullcline in the topology of the attractor.

AbstractWhen a laser system is studied, the time evolution of the light intensity emitted by the laser cavity is recorded. The phase portrait reconstructed from that time series never presents symmetry although the amplitude equations always generate phase portraits with symmetry properties. It is shown that the detuning between the normalized steady-state laser frequency and the molecular resonance frequency induces a continuous rotation symmetry. This equivariant dynamics is linked with the dynamics underlying the experimental observations of the light intensity through a system for which the symmetry properties are modded out.

Abstract
Obtaining a global model from the z-variable of the Rössler system is considered to be difficult because of its spiky structure. In this Letter, a 3D global model from the z-variable is derived in a space spanned by the state variable of the time-series itself
and generic functions of the other two state variables.We term this space the Ansatz Space. The procedure consists of two steps. First, models built in the derivative coordinates are obtained. Second, we use the analytical form of the map ϕ between systems in the original state space and in the differential space to find a class of models in the Ansatz Space. We find eight models in this class which we show to be dynamically equivalent to the original Rössler system. The important attribute of this approach is that we do not need to use any prior knowledge of the dynamical system other than the measured time series data in order to obtain global models from a single time series.

AbstractA chaotic attractor with symmetry group G can be mapped down to an image chaotic attractor without symmetry by a smooth mapping with singularities. The image chaotic attractor can be lifted to many distinct structurally stable strange attractors, each equivariant under G, all with the same image chaotic attractor. If the symbolic dynamics of the image chaotic attractor requires s symbols σ1,σ2,…,σs, then |G|s symbols are required for symbolic dynamics in the covers, and there are |G|s distinct equivariant covers. The covers are distinguished by an index. The index is an assignment of a group operator to each symbol σi:σi→gαi. The subgroup H⊂G generated by the group operators gαi in the index determines how many disconnected components (|G|/|H|) each equivariant cover has. The components are labeled by coset representatives from G/H. The structure of each connected component is determined by H. A simple algorithm is presented for determining the number and the period of orbits in an equivariant attractor that cover an orbit of period p in the image attractor. Modifications of these results for structurally unstable covers are summarized by an adjacency diagram.

AbtractThis paper investigates nonlinear wave–wave interactions in a system that describes a modified decay instability and consists of three Langmuir and one ion-sound waves. As a means to establish that the underlying dynamics exists in a 3D space and that it is of the Lorenz-type, both continuous and discrete-time multivariable global models were obtained from data. These data were obtained from a 10D dynamical system that describes the modified decay instability obtained from Zakharov’s equations which characterise Langmuir turbulence. This 10D model is equivariant under a continuous rotation symmetry and a discrete order-2 rotation symmetry. When the continuous rotation symmetry is modded out, that is, when the dynamics are represented with the continuous rotation symmetry removed under a local diffeomorphism, it is shown that a 3D system may describe the underlying dynamics. For certain parameter values, the models, obtained using global modelling techniques from three time series from the 10D dynamics with the continuous rotation symmetry modded out, generate attractors which are topologically equivalent. These models can be simulated easily and, due to their simplicity, are amenable for analysis of the original dynamics after symmetries have been modded out. Moreover, it is shown that all of these attractors are topologically equivalent to an attractor generated by the well-known Lorenz system.